Challenges and Possibilities in Teaching and Learning of Calculus : A Case Study of India

Abstract

Introducing, and thereafter teaching calculus to senior secondary, early college and university students, at the expense of algebra and geometry, is causing half-baked calculus being served to relatively under-prepared students. In line with this proposition, the current research aims to identify how cognition of calculus takes place among learners, what teaching methodologies are used by Indian teachers, what pedagogical techniques are most efficient in calculus teaching, and what prerequisites are called for before commencement of the course on calculus? For this extensive study, data was gathered from school teachers and assistant/associate professors of colleges and universities, having more than 6 years of calculus teaching experience, drawn from 26 schools, 19 colleges and 7 university departments, spanning across 23 different states and union territories of India. A total of 142 teachers took part in this study. Data was collected using schedules, classroom observations, focus group interviews, and informal discussions that were carried out both before and after the classroom teaching. NVivo and Concordance softwares were used for analysis of the emerging content and classroom discourses. The study traversing between February 2016 to April 2019, is qualitative in its framework and lies purely within the interpretivist paradigm. The findings of this research shall mellow the understanding of calculus cognition operational among school, college and university going students.

Keywords

• Calculus,Classroom,Curriculum,Differentiation,Evaluation,Integration,Learning,Mathematics Education

References

1. Armstrong, G., Garner, L., & Wynn, J. (1994). Our experience with two reformed calculus programs. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 4(4), 301-311.
2. Artigue, M. (2001). What can we learn from educational research at the university level? The teaching and learning of mathematics at university level (pp. 207-220): Springer.
3. Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. E. (1997). The development of students' graphical understanding of the derivative. The Journal of Mathematical Behavior, 16(4), 399-431.
4. Bell, A., & Janvier, C. (1981). The interpretation of graphs representing situations. For the learning of mathematics, 2(1), 34-42.
5. Bonsangue, M. V., & Drew, D. E. (1995). Increasing minority students' success in calculus. New Directions for Teaching and Learning, 1995(61), 23-33.
6. Bookman, J. (1993). An expert novice study of metacognitive behavior in four types of mathematics problems. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 3(3), 284-314.
7. Bookman, J., & Blake, L. (1996). Seven years of Project CALC at Duke University approaching steady state? Problems, Resources, and Issues in Mathematics Undergraduate Studies, 6(3), 221-234.
8. Bookman, J., & Friedman, C. P. (1994). A comparison of the problem solving performance of students in lab based and traditional calculus. Research in collegiate mathematics education I, 4, 101-116.

9. Bowers, D. (1999). Animating web pages with the TI-92. Retrieved July.
10. Brunsell, E., & Horejsi, M. (2013). Flipping Your Classroom in One" Take". The Science Teacher, 80(3), 8.
11. Buck, R. (1970). A generalized Hausdorff dimension for functions and sets. Pacific Journal of Mathematics, 33(1), 69-78.
12. Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., & Perlwitz, M. (1991). Assessment of a problem-centered second-grade mathematics project. Journal for research in mathematics education, 3-29.
13. Confrey, J. (1981). Conceptual Change, Number Concepts And The Introduction To Calculus.
14. Dada, O., & Akpan, S. M. Discriminant Analysis of Psycho-Social Predictors of Mathematics Achievement of Gifted Students in Nigeria. Journal for the Education of Gifted Young Scientists, 7(3), 581-594.
15. David, H. (2014). Israel’s Achievements in Mathematics in the Last International Examinations: Part I: The TIMSS 2011. Journal for the Education of Gifted Young Scientists, 2(1), 11-17.
16. Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. The Journal of Mathematical Behavior.
17. De Guzmán, M., Hodgson, B. R., Robert, A., & Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. Paper presented at the Proceedings of the international Congress of Mathematicians.
18. Demana, F. D., Waits, B. K., & Clemens, S. R. (1993). Precalculus: Functions and graphs: Addison Wesley.
19. Dossey, J. A. (1992). The nature of mathematics: Its role and its influence. Handbook of research on mathematics teaching and learning, 39, 48.
20. Douglas, R. G. (1995). The first decade of calculus reform. UME trends, 6(6), 1-2.
21. Dreyfus, T., & Eisenberg, T. (1983). The function concept in college students: Linearity, smoothness and periodicity. Focus on learning problems in mathematics, 5(3), 119-132.
22. Dreyfus, T., & Eisenberg, T. (1984). Intuitions on functions. The Journal of experimental education, 52(2), 77-85.
23. Dreyfus, T., & Eisenberg, T. (2012). On different facets of mathematical thinking The nature of mathematical thinking (pp. 269-300): Routledge.
24. Dubinsky, E. (2002). Reflective abstraction in advanced mathematical thinking Advanced mathematical thinking (pp. 95-126): Springer.
25. Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. MAA notes, 31-46.
26. Ferrini-Mundy, J., & Graham, K. G. (1991). An overview of the calculus curriculum reform effort: Issues for learning, teaching, and curriculum development. The American Mathematical Monthly, 98(7), 627-635.
27. Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and learning calculus. Research ideas for the classroom: High school mathematics, 155-176.
28. Fulton, K. (2012). Upside down and inside out: Flip your classroom to improve student learning. Learning & Leading with Technology, 39(8), 12-17.
29. Gardiner, T. (1995). Mathematics hamstrung by long divisions. The Sunday Times, 22.
30. Garofalo, J. (1989). Beliefs and their influence on mathematical performance. The Mathematics Teacher, 82(7), 502-505.
31. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A" proceptual" view of simple arithmetic. Journal for research in mathematics education, 116-140.
32. Hartinah, S., Suherman, S., Syazali, M., Efendi, H., Junaidi, R., Jermsittiparsert, K., & Umam, R. Probing-Prompting Based On Ethnomathematics Learning Model: The Effect On Mathematical Communication Skill. Journal for the Education of Gifted Young Scientists, 7(4), 799-814.
33. Herreid, C. F., & Schiller, N. A. (2013). Case studies and the flipped classroom. Journal of College Science Teaching, 42(5), 62-66.
34. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics, 65-97.
35. Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. The Journal of Mathematical Behavior, 17(2), 265-281. Kay, R., & Kletskin, I. (2012). Evaluating the use of problem-based video podcasts to teach mathematics in higher education. Computers & Education, 59(2), 619-627.
36. Kennedy, D. (2000). AP calculus for a new century: Consultado el.
37. Koirala, H. P. (1997). Teaching of calculus for students’ conceptual understanding. The Mathematics Educator, 2(1), 52-62.
38. Kök, B., & Davaslıgil, Ü. (2014). The effect of teaching geometry which is differentiated based on the parallel curriculum for gifted/talented students on spatial ability. Journal for the Education of Gifted Young Scientists, 2(1), 40-52.
39. Kuh, G. D., Kinzie, J., Schuh, J. H., & Whitt, E. J. (2011). Student success in college: Creating conditions that matter: John Wiley & Sons.
40. Legrand, M. (1993). Débat scientifique en cours de mathématiques. Repères irem, 10, 123-159.
41. Leinbach, C. (1997). The curriculum in the age of CAS. The state of computer algebra in mathematics education. Bromley, England: Chartwell-Bratt.
42. Markovits, Z., Eylon, B.-S., & Bruckheimer, M. (1986). Functions today and yesterday. For the learning of mathematics, 6(2), 18-28. Mathematics, N. C. o. T. o. M. C. o. S. f. S. (1989). Curriculum and evaluation standards for school mathematics: Natl Council of Teachers of.
43. Meadows, M. (2016). Where Are All the Talented Girls? How Can We Help Them Achieve in Science Technology Engineering and Mathematics? Journal for the Education of Gifted Young Scientists, 4(2), 29-42.
44. Monk, D. H. (1987). Secondary school size and curriculum comprehensiveness. Economics of Education Review, 6(2), 137-150.
45. Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. CBMS Issues in Mathematics Education, 4, 139-168.
46. Neumark, V. (1995). For the love of maths. Times Educational Supplement, 8.
47. Noddings, N., Maher, C. A., & Davis, R. B. (1990). Constructivist views on the teaching and learning of mathematics : National Council of Teachers of Mathematics.
48. Orton, A., & Wain, G. (1994). The aims of teaching mathematics. Issues in teaching mathematics, 1-20.
49. Palmiter, J. R. (1991). Effects of computer algebra systems on concept and skill acquisition in calculus. Journal for research in mathematics education, 151-156.
50. Pardimin, P., Arcana, N., & Supriadi, D. Developing Media Based on the Information and Communications Technology to Improve The Effectiveness of The Direct Instruction Method in Mathematics Learning. Journal for the Education of Gifted Young Scientists, 7(4), 1311-1323.
51. Park, K., & Travers, K. J. (1996). A comparative study of a computer-based and a standard college first-year calculus course. CBMS Issues in Mathematics Education, 6, 155-176.
52. Piaget, J. (1985). The equilibration of cognitive structures (T. Brown & KJ Thampy, Trans.). Cambridge: Harvard University Presss.
53. Praslon, F. (1999). Discontinuities regarding the secondary/university transition: The notion of derivative as a specific case. Paper presented at the PME conference.
54. Repo, S. (1994). Understanding and reflective abstraction: Learning the concept of derivative in a computer environment. International DERIVE Journal, 1(1), 97-113.
55. Ruddick, K. W. (2012). Improving chemical education from high school to college using a more hands-on approach: The University of Memphis.
56. Schnepp, M., & Nemirovsky, R. (2001). Constructing a foundation for the fundamental theorem of calculus. The roles of representation in school mathematics, 90-102.
57. Schoenfeld, A. H. (1995). A brief biography of calculus reform. UME trends, 6(6), 3-5.
58. Selden, J., Selden, A., & Mason, A. (1994). Even good calculus students can't solve nonroutine problems. MAA notes, 19-28.
59. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification-the case of function. The concept of function: Aspects of epistemology and pedagogy, 25, 59-84.
60. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics teaching, 77(1), 20-26.
61. Smith, D. A., & Moore, L. C. (1990). Duke university: Project calc. Priming the calculus pump: Innovations and resources, 51-74.
62. Smith, D. A., & Moore, L. C. (1991). Project CALC: An integrated laboratory course. The laboratory approach to teaching calculus. The Mathematical Association of America, Washington, DC, 81-92.
63. Solow, A. E. (1994). Preparing for a new calculus: Conference proceedings: Mathematical Assn of Amer.
64. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. Handbook of research on mathematics teaching and learning, 495-511.
65. Tall, D. (1996). Functions and Calculus (Vol. 1): Dordrecht, Netherlands: Kluwer Academic.
66. Tall, D., & Blackett, N. (1986). Investigating graphs and the calculus in the sixth form. Exploring mathematics with microcomputers, 156-175.
67. Tall, D., & Schwarzenberger, R. (1978). Conflicts in the learning of real numbers and limits. Mathematics teaching, 82.
68. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in mathematics, 12(2), 151-169.
69. Tortop, H. S. (2014). Examining the effectiveness of the in-service training program for the education of the academically gifted students in Turkey: A case study. Journal for the Education of Gifted Young Scientists, 2(2), 67-86.
70. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for research in mathematics education, 356-366.
71. White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for research in mathematics education, 27, 79-95.
72. Williams, S. R. (1991). Models of limit held by college calculus students. Journal for research in mathematics education, 22(3), 219-236.
73. Young, G. S. (1987). Present problems and future prospects. Calculus for a new century, 172-175.
74. Zappe, S., Leicht, R., Messner, J., Litzinger, T., & Lee, H. W. (2009). " Flipping" the classroom to explore active learning in a large undergraduate course. Paper presented at the ASEE Annual Conference and Exposition, Conference Proceedings.